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in

(

t >

0)

×

Ω

, with the initial conditions

u

(0

, x

) =

f

(

x

);

u

t

(0

, x

) =

g

(

x

)

for

x

Ω

(9)

and the boundary condition

u

= 0

(10)

on

(

t >

0)

×

Γ

. We assume the initial functions

f

o

H

1

(Ω)

and

g

L

2

(Ω)

are real-valued.

It is well known that a solution of the problem (8)–(10) satisfies the

energy conservation law

E

(

t

) =

k

u

t

k

2

L

2

(Ω)

+

k∇

u

k

2

L

2

(Ω)

=

E

(0)

(11)

for

t >

0

.

Below we study the connections between spectral properties of the

Laplace operator and the behavior for large values of time to solutions

of the problem (8)–(10). We investigate stabilization in time with special

regard to the properties of the local energy function

E

Ω

0

(

t

) =

k

u

t

k

2

L

2

0

)

+

k∇

u

k

2

L

2

0

)

(12)

where

Ω

0

Ω

is a bounded domain.

Let us note that many practically important problems deal with

unbounded domains

Ω

so the inverse operator

L

1

may be unbounded

in the main space

H

=

L

2

(Ω)

.

Solutions from the Energy Class and Spectral Representation.

We

consider solutions of the problem (8)–(10) from the energy class (see

[11]), that is a function

u

(

t, x

)

C

([0

,

+

);

o

H

1

(Ω))

such that

u

t

(

t, x

)

C

([0

,

+

);

L

2

(Ω))

satisfying the initial conditions (9), the equality (11)

and the integral identity

T

Z

0

Z

Ω

((

u,

w

)

u

t

w

t

)

dt dx

=

Z

Ω

g

(

x

)

w

(0

, x

)

dx,

for all

w

H

1

((0

, T

)

×

Ω)

satisfying

w

= 0

on

(0

, T

)

×

Γ

and

w

(

T, x

) = 0

,

T >

0

.

Consider a self-adjoint non-negative operator

L

:

D

(

L

)

L

2

(Ω)

generated in

L

2

(Ω)

space by differential expression

Δ

with Dirichlet

boundary condition. Using an estimates to solutions of elliptic boundary

value problems [14], we have the domain of operator

L

:

D

(

L

) =

o

H

1

(Ω)

H

2

0

)

∩ {

Δ

u

L

2

(Ω)

}

,

here

Ω

0

Ω

is an arbitrary bounded domain.

Let

{

E

(

λ

)

}

,

−∞

< λ <

+

, be a family of spectral projectors associated

with operator

L

[15].

6

ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 3